The Z-score cannot be interpreted so easily. It must be evaluated over all the different ways people have tried to test the digits for randomness.
"Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found." - https://en.wikipedia.org/wiki/Pi
There's a large literature of people looking for non-randomness in pi, and failing.
If you try 50,000 different tests, with a z-score of 3.29 then 0.1% - 50 of them - will give false positives.
Otherwise you risk being seen as yet another math crank.
If you'll forgive the reductio ad absurdem: I could toss a coin once and post a misleading write up about how coin tosses land on heads 100% of the time. :)
Also notable: the description of your repo (at time of writing) is literally "Statistically significant proof that the digits of pi are not random". The word "proof" there does seem to contradict what you're saying in this message.
Not bullying by the way - I'm well familiar with how exciting it can be to find something cool. (And kudos for sharing)
But if you use terms like "statistically significant proof" then you're making claims / misleading statements (or just being clickbaity) rather than providing objective observations and _separare_ subjective hypetheses.
(IMO) a far more palatable alternative would be "I tried analysing the first N digits of pi with a statistical algorithm they're not evenly distributed." (ie. Just the facts, no claims) or if you want to retain the clickbait, "I used statistics to analyse the first N digits of pi.. you won't believe the 9,000th digit!"
As others have mentioned here: of course it's not random. We can predict the next digit with 100% accuracy. But even meeting you halfway and choosing to interpret your use of "random" as meaning "evenly distributed" it's a big reach
The only thing one can conclude from your claim is that, the first handful number of digits in pi can be very, very slightly compressed if you are somehow able to ignore the size of that classifier [1]. The algorithmic randomness however requires the classifier to be included, and there is no known self-contained program that is smaller than printing all digits in verbatim or computing them in the first place.
[1] By the way, this is not really surprising at all. In fact there are 5,001 (not 5,000) out of first 10,000 decimal digits of pi that are between 5 and 9, so it can be very, very slightly compressed in that way---of course after the decompressor excluded.
The second problem is the interpretation of statistical significance. You don't even explain how you compute it, and it's not in the code. For starters, what's your df?
Another problem is the sample. It might be possible that any range of 10,000 digits isn't random (according to your criterion), but the whole is. Now that I think about it: it is very likely there is a structure, since it's a rather limited string, and the number of possible bit strings of length n quickly outnumbers the total length of the string for increasing n. 10k bits can only contain the unique strings up to length 10, and those wouldn't be random at all (since you can predict the remaining bits with increasing accuracy). So the details of your prediction model are really essential.
Pi is irrational but definitely not random.
